Question: What is the greatest common factor of $30b^{2}$, $80b^{2}$, and $20b^{3}$ ?
Answer: Let's factor each monomial to its prime factors: $\begin{aligned} 30b^{2}&=(2)(3)(5)(b)(b) \\\\ 80b^{2}&=(2)(2)(2)(2)(5)(b)(b) \\\\ 20b^{3}&=(2)(2)(5)(b)(b)(b) \end{aligned}$ We want the largest set of factors that's included in all three monomials. All of the monomials have one factor of $ 2$, one factor of $ 5$, and two factors of $ b$ : $\begin{aligned} 30b^{2}&=( 2)(3)( 5)( b)( b) \\\\ 80b^{2}&=( 2)(2)(2)(2)( 5)( b)( b) \\\\ 20b^{3}&=( 2)(2)( 5)( b)( b)(b) \end{aligned}$ This is the greatest common factor: $( 2)( 5)( b)( b)=10b^{2}$